Quasirandom permutations are characterized by 4-point densities

For permutations π and τ of lengths |π| ≤ |τ |, let t(π, τ) be the probability that the restriction of τ to a random |π|-point set is (order) isomorphic to π. We show that every sequence {τj} of permutations such that |τj| → ∞ and t(π, τj) → 1/4! for every 4-point permutation π is quasirandom (that is, t(π, τj) → 1/|π|! for every π). This answers a question posed by Graham.